Factor the following expression: $-7$ $x^2$ $-23$ $x+$ $20$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(20)} &=& -140 \\ {a} + {b} &=& & & {-23} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-140$ and add them together. Remember, since $-140$ is negative, one of the factors must be negative. The factors that add up to ${-23}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${-28}$ $ \begin{eqnarray} {ab} &=& ({5})({-28}) &=& -140 \\ {a} + {b} &=& {5} + {-28} &=& -23 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 +{5}x {-28}x +{20} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 +{5}x) + ({-28}x +{20}) $ Factor out the common factors: $ x(-7x + 5) + 4(-7x + 5) $ Notice how $(-7x + 5)$ has become a common factor. Factor this out to find the answer. $(-7x + 5)(x + 4)$